It is used for all of the above. In essence it is a statement along the lines of ##F=\sum ma## for a continuous fluid. Typically velocity and pressure are the most desirable quantities, but along with the energy equation, temperature and heat transfer are commonly sought, too.

So basically, in vector notation, you have
[tex]\rho \dfrac{D\vec{V}}{Dt} = -\nabla p + \mu\nabla^2 \vec{V} + \vec{f}.[/tex]
The terms are basically force terms, where
[tex]\rho\dfrac{D\vec{V}}{Dt} = \rho \vec{a}[/tex]
is the total force per unit volume on a given fluid element on the left side, and the three terms on the right side area as follows:

##-\nabla p## are the pressure forces per unit volume on the fluid element;

##\mu\nabla^2 \vec{V}## are the viscous forces per unit volume on the fluid element; and

##\vec{f}## is the body force per unit volume on the fluid element (e.g. gravity or magneto/electrodynamic forces).

From what I have seen in fluids calculations they're either based on the Bernoulli equation, or derived almost entirely experimentally, with the addition of friction factors added in. This is for internal flow, by the way.

Perhpas it would help to understand what they are NOT used for. They are not used for 1 dimensional flow such as the flow and pressure drop of fluid through a pipe. For that, the Darcy Weisbach equation and similar equations are generally used. The NS equations COULD be used for that but in general, the equations can't be solved for any but the most simple systems and any real piping system would be far too complex.

The NS equations are generally used by computers running CFD programs. The flow field is discretized such as by reducing to finite volumes or elements, and the equations are then linearized to allow them to be solved algebraically. The use of CFD software is generally limited to fairly complex fluid flows where 2 or 3 dimensional flow needs to be analyzed such as flow through a turbine for example or across an aircraft. Flow through valves and piping is rarely analyzed using CFD.

Was just curious, as I never actually dealt with the NS equation during my undergrad and was wondering if I was "missing out" on anything. The fluids taught to me was mainly internal flow, where we made use of the Bernoulli equation, the Darcy–Weisbach equation, alongside all sorts of other experimentally derived equations to calculate liquid and gas flow through piping, valves, instrumentation etc.